On the nonconnectedness of moduli spaces of arrangements, II: construction of nonarithmetic pairs

TL;DR

This paper introduces algorithms to construct nonarithmetic pairs of line arrangements that are lattice isomorphic but not lattice isotopic, expanding understanding of moduli space structures in arrangement theory.

Contribution

The paper develops two algorithms for constructing nonarithmetic pairs of arrangements over different fields, including explicit examples in complex, real, and rational cases.

Findings

Algorithms successfully generate nonarithmetic pairs

Explicit examples demonstrate the algorithms' effectiveness

New nonarithmetic pairs over various fields are constructed

Abstract

Constructing lattice isomorphic line arrangements that are not lattice isotopic is a complex yet fundamental task. In this paper, we focus on such pairs but which are not Galois conjugated, referred to as nonarithmetic pairs. Splitting polygons have been introduced by the author to facilitate the construction of lattice isomorphic arrangements that are not lattice isotopic. Exploiting this structure, we develop two algorithms which produce nonarithmetic pairs: the first generates pairs over a number field, while the second yields pairs over the rationals. Moreover, explicit applications of these algorithms are presented, including one complex, one real, and one rational nonarithmetic pair.